1984 Reports

Linear Problems (with Extended Range) Have Linear Optimal Algorithms

Packel, Edward W.

Abstract. Let F1 and F2 be normed linear spaces and S: F0→F2 a linear operator on a balanced subset F0 of F2. If N denotes a finite dimensional linear information operator on F0, it is known that there need not be a linear algorithm φ:N(F0)→F2 which is optimal in the sense that ||φ:(N(f))-S(f)|| is minimized. We show that the linear problem defined by S and N can be regarded as having a linear optimal algorithm if we allow the range of φ to be extended in a natural way. The result depends upon imbedding F2 isometrically in the space of continuous functions on a compact Hausdorff space X. This is done by making use of a consequence of the classical Banach-Alaoglu theorem.